Once upon a time, before there was acrylic, crochet was done with tiny hooks the size of sewing needles with cotton thread. It was very lacy and was used to make decorative collars, covered coin purses and elaborate tablecloths. Here’s an example of filet work by my great grandmother.
Some people still crochet this way, but it’s painstakingly slow and those stitches are tiny. Here’s a coin purse in progress by a friend of mine that I photographed over a year ago. I’m pretty sure it isn’t finished yet.
Things got a little more fun once more colorful dyes became available, but it was still mostly tiny lace. My great aunt made this doily.
Most people imagine the acrylic granny square afghans which were so popular in the 1970s when they think of crochet. There were some ugly, itchy blankets, but I blame the yarn, not the technique. Here’s a lovely granny square afghan a friend made. 
My first crocheted afghan was a collaboration with my mother. She started this project, realized that it was a bigger task than she anticipated and enlisted me to help her finish it. I crocheted half of the hexagons and we sewed them all together and wove in a ridiculous number of ends. It was a massive undertaking even for two people with years of experience. This is a message to all the overenthusiastic new cr0cheters who want to make a king sized blanket for their first project. Do yourself a favor and start with a hot pad or a pillow cover.
My favorite crochet projects are shawls made in lightweight, drapey fibers. This one is made in a cotton/silk blend and is based on the very popular virus shawl, which is a free pattern. You will need to scroll down a little to upload this pattern.
The next shawl is one of my all time favorites. The yarn is a wool/llama blend, so it’s super cozy as well as colorful. The pattern is Lost In Time, and yes, it’s another free pattern.
This year my challenge was to make a flattering crochet cardigan. Crochet can be thicker and less drapey than knit material. My solution was to do the rabbit hole cardigan, which is knit similar to a granny square, in a silk/alpaca blend. This pattern is basically a giant granny square, but the super soft, drapey yarn couldn’t look less like grandma’s afghan.
I tried this pattern again, this time pairing a fuzzy mohair with lace weight alpaca yarn. I crocheted this with a V stitch instead of shell stitch. It’s quite warm in spite of the open, lacy structure. 
So there’s a tour of fancy crochet by my friends, family and myself. In spite of the intimidating examples I’ve shown here, most people agree that it is easier to learn than knitting because you only have to manage one loop at a time. And if you ever see something in a store that you recognize as crochet, remember that somebody made it by hand. There are no machines that can crochet.